Isomorphism of AW ∗ -Algebras

Abstract

H. A. Dye, in his paper On the geometry of projections in certain operator algebras,' has shown that the structure of the projection lattice of an AW*-algebra determines the structure of the algebra. He then goes on to show that, in the case of a factor not of type 12n, the structure of projection lattice is determined by the unitary structure. The crucial step in his proof lies in the study of the family of unitary operators p (Xe +1e) where e is a fixed projection and X takes on all complex numbers of absolute value 1. Following his method by examining the operators p(Xe + 1 e) in greater detail we are able to extend his result. However, since we rely heavily on the existence of four orthogonal equivalent projections, we are not able to get any information about algebras of type I2n+l1 We first collect a few results about AW*-algebras which are pertinent to our discussion. (a) If e, f are orthogonal equivalent projections then there exists a unitary operator u such that f=ueu-1 and u(1 -e-f) = (1 -e-f)u 1 e -f. In fact, if v is a partial isometry with vv =e, v* =f, then u=v+v*+(1-e-f) will do. (b) A projection e is called properly infinite if, for every central projection g, ge is either infinite or zero. Given any projection e there is a central projection g such that ge is properly infinite (or zero) and (1 -g)e is finite. (c) By the central cover of a projection e we mean the least central projection containing e. Let e be properly infinite and f finite. Then, if e, f have the same central cover, e contains infinitely many orthogonal projections equivalent to f. Let M, N be AW*-algebras having no component of type Inv p an algebraic isomorphism between the unitary groups M. and N.. De-